Simplicial lists in operad theory I
Redi Haderi, \"Ozg\"un \"Unl\"u

TL;DR
This paper introduces a new categorical framework using simplicial lists to model non-symmetric operads and their higher-dimensional generalizations, providing new tools for operad theory and $ ext{infinity}$-operads.
Contribution
It defines the category of simplicial lists, constructs a nerve functor for operads, and develops a model for non-symmetric $ ext{infinity}$-operads, expanding operad theory tools.
Findings
Constructed a fully-faithful nerve functor from operads to simplicial lists.
Showed that the category of simplicial lists is a presheaf category.
Developed a coherent nerve functor for $ ext{infinity}$-operads.
Abstract
We define a category whose objects are sets and morphisms are mappings which assign to an element in the domain an ordered sequence (list) of elements in the codomain. We introduce and study a category of simplicial objects whose objects are functors , which we call simplicial lists, and morphisms are natural transformations which have functions as components. We demonstrate that supports the combinatorics of (non-symmetric) operads by constructing a fully-faithful nerve functor from the category of operads. This leads to a reasonable model for the theory of non-symmetric -operads. We also demonstrate that has the structure of a presheaf category. In particular, we study a subcategory of operadic simplicial…
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Taxonomy
TopicsAdvanced Topics in Algebra · Homotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models
